reserve n for Nat;

theorem
  for f be FinSequence of TOP-REAL 2 for p be Point of TOP-REAL 2 st p
  in L~f holds L_Cut (f,p) <> {}
proof
  let f be FinSequence of TOP-REAL 2;
  let p be Point of TOP-REAL 2;
  assume that
A1: p in L~f and
A2: L_Cut (f,p) = {};
  len f <> 0 by A1,TOPREAL1:22;
  then f <> {};
  then
A3: len f in dom f by FINSEQ_5:6;
A4: p<>f.(Index(p,f)+1) or p=f.(Index(p,f)+1);
  <*p*>^mid(f,Index(p,f)+1,len f) is non empty;
  then L_Cut(f,p) = mid(f,Index(p,f)+1,len f) by A2,A4,JORDAN3:def 3;
  then not Index(p,f)+1 in dom f by A2,A3,SPRECT_2:7;
  then Index(p,f)+1 < 1 or Index(p,f)+1 > len f by FINSEQ_3:25;
  then Index(p,f) >= len f by NAT_1:11,13;
  hence contradiction by A1,JORDAN3:8;
end;
